Finite determinacy and Whitney equisingularity of map germs from C(n) to C(2n-1)

We show that a holomorphic map germ f : (C(n), 0) -> (C(2n-1), 0) is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n >= 3, we have that mu(D(2)(f)) = 2 mu(D(2)(f)/S(2))+C(f)-1, where D(2)(f) is the lifting of the double point curve in (C(n) x C(n), 0), mu(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f(t)(x)) of f and show that if F is mu-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f(t) is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular.

M-deformations of A-simple germs from R(n) to R(n+1)

All A-simple corank-1 germs from R(n) to R(n+1), where n not equal 4, have an M-deformation, that is a deformation in which the maximal numbers of isolated stable singular points are simultaneously present in the image.

A note on finite determinacy for corank 2 map germs from surfaces to 3-space

We study properties of finitely determined corank 2 quasihomogeneous map germs f: (C(2), 0) -> (C(3), 0). Examples and counter examples of such map germs are presented.

Real map germs and higher open book structures

The paper focusses on the existence of higher open book structures defined by real map germs psi : (R(m), 0) -> (R(p), 0) such that Sing psi boolean AND psi(-1)(0) subset of {0}. A general existence criterion is proved, with view to weighted-homogeneous maps.

Lapanthus (Bromeliaceae, Bromelioideae): A New Genus from the Southern Espinhaco Range, Brazil

In this paper we describe a new genus of Bromehaceae, Lapanthus, restricted to the southern portion of the Espinhaco Range, Minas Germs State, in southeastern Brazil Two new combinations to accommodate species previously described in the genera Orthophytum and cryptanthus and one new synonym are proposed Lapanthus has morphological affinities with both Cryptanthus and Orthophytum but nevertheless differs by the combination of margins of the petals ciliate, presence of lanceolate petal appendages and free stamens, and also by molecular data Cryptanthus and Orthophytum have petals entire along the margins, and the filaments of the most internal whorl are adnate to the petals Lapanthus stands out by having a pair of lanceolate petal appendages, which are almost completely adnate to the petals In Orthophytum, however, appendages are cupuhform or sacciform and they are totally absent in the genus Cryptanthus Lapanthus and Orthophytum present meiotic and mitotic chromosome numbers equal to n=25 and 2n=50, 100 and 150 respectively, while Cryptanthus presents meiotic and mitotic chromosome numbers n=17 and 2n=34, 36, 54 respectively, and this difference is considered to be an autapomorphic feature of Cryptanthus Descriptions of the genus and species, identification keys, illustrations, photographs of living specimens, and taxonomic comments are provided

GTPases RhoA and Rac1 are important for amelogenin and DSPP expression during differentiation of ameloblasts and odontoblasts

Morphogenesis and cytodifferentiation are distinct processes in tooth development. Cell proliferation predominates in morphogenesis; differentiation involves changes in form and gene expression. The cytoskeleton is essential for both processes, being regulated by Rho GTPases. The aim of this study was to verify the expression, distribution, and role of Rho GTPases in ameloblasts and odontoblasts during tooth development in correlation with actin and tubulin arrangements and amelogenin and dentin sialophosphoprotein (DSPP) expression. RhoA, Rac1, and Cdc42 were strongly expressed during morphogenesis; during cytodifferentiation, RhoA was present in ameloblasts and odontoblasts, Rac1 and its effector Pak3 were observed in ameloblasts; and Cdc42 was present in all cells of the tooth germ and mesenchyme. The expression of RhoA mRNA and its effectors RockI and RockII, Rac1 and Pak3, as analyzed by real-time polymerase chain reaction, increased after ameloblast and odontoblast differentiation, according to the mRNA expression of amelogenin and DSPP. The inhibition of all Rho GTPases by Clostridium difficile toxin A completely abolished amelogenin and DSPP expression in tooth germs cultured in anterior eye chamber, whereas the specific inhibition of the Rocks showed only a partial effect. Thus, both GTPases are important during tooth morphogenesis. During cytodifferentiation, Rho proteins are essential for the complete differentiation of ameloblasts and odontoblasts by regulating the expression of amelogenin and DSPP. RhoA and its effector RockI contribute to this role. A specific function for Rac1 in ameloblasts remains to be elucidated; its punctate distribution indicates its possible role in exocytosis/endocytosis.

C(0) and bi-Lipschitz K-equivalence of mappings

In this paper we investigate the classification of mappings up to K-equivalence. We give several results of this type. We study semialgebraic deformations up to semialgebraic C(0) K-equivalence and bi-Lipschitz K-equivalence. We give an algebraic criterion for bi-Lipschitz K-triviality in terms of semi-integral closure (Theorem 3.5). We also give a new proof of a result of Nishimura: we show that two germs of smooth mappings f, g : R(n) -> R(n), finitely determined with respect to K-equivalence are C(0)-K-equivalent if and only if they have the same degree in absolute value.

The doodle of a finitely determined map germ from R(2) to R(3)

Let f : U subset of R(2) -> R(3) be a representative of a finitely determined map germ f : (R(2), 0) -> (R(3), 0). Consider the curve obtained as the intersection of the image of the mapping f with a sufficiently small sphere s(epsilon)(2) centered at the origin in R(3), call this curve the associated doodle of the map germ f. For a large class of map germs the associated doodle has many transversal self-intersections. The topological classification of such map germs is considered from the point of view of the associated doodles. (C) 2009 Elsevier Inc. All rights reserved.